Start: 19 Mar 2018.
Last update: 28 Oct 2020
Max. nonlinear number of knots = 1 (df = 2).
Using cases/N for all sex/age classes.
Statistical comparison of models composed of different combinations of qualitatively chosen variables.
Main goal: assess relative importance of intrinsic (susceptibles) vs extrinsic (environment) drivers.
Approach:
- Comparison of GAMs using a combination of “feature selection” and adjusted R^2.
- (add likelihood ratio tests?) - Random Forests or Boosted Regression Trees - Hierarchical Bayesian model will be attempted. If successful, this would give more refined information about where (susceptibles, transmission/contacts, susceptibility) environment acts most strongly.
Model comparison of GAMS is done using hv-block crossvalidation, for models containing up to 4 variables, assessing all combinations.
Model stats are:
- Adjusted R2
- Feature mismatching (number of mismatching outbreak categories).
Ranking is done based on the combined ranks of these two statistics:
Each model is ranked according to each statistic, resulting in 2 ranking values where 1 is best.
These values are summed for each model, and this combined value is used for final ranking.
Ties receive the same ranking value, and the next ranking model receives the next ranking value.
E.g. if 4 models have the same ranking, they als receive value 1. Model 5, the next best ranking, then gets value 2 (as opposed to value 5 because it is the fifth model). This ensures that small differences between values are not penalized too much.
Before ranking, all values are rounded to two digits behind the comma (or whatever this is called).
Lasso regression, random forests and boosted regression trees are used as a parallel approach, to see whether the same variables are selected.
Dataset:
- Cases without male subadult/adults
- Susceptible reconstruction, scale 100, yearlings (1y old) + male juveniles (2+3y old)
- SST south, months 6-7, 8-10 (means)
- SST central, months 6-7 and 8-10 (means)
- SST north, months 8-10 (mean)
- Spring transition at 36N
- Spring transition at 39N
- Spring transition at 45N
- Upwelling south (36N), months 8-10 (mean)
- Upwelling central (39N), months 8-10 (mean)
- Pup survival as proxy for conditions in south - Yearling survival as proxy for conditions in south & center
| variable | dfs |
|---|---|
| Srec.yearlings.male.juv | 1 |
| Pup.surv | 1 |
| Yearling.surv | 1 |
| ST36 | 1 |
| ST39 | 1 |
| ST45 | 1 |
| SSTsouth.6to7 | 1 |
| SSTsouth.8to10 | 1 |
| SSTcentral.6to7 | 1 |
| SSTcentral.8to10 | 1 |
| SSTnorth.8to10 | 1 |
| Upw.south.8to10 | 2 |
| Upw.central.8to10 | 1 |
| parms | Combined.rank | AdjR2.cv | AdjR2.full | FMM.cv | FMM.full | AIC.full | deltaAIC | Pval | AdjR2.rank | FMM.rank | prop.dev.explained |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Srec.yearlings.male.juv - ST36 - SSTcentral.6to7 - Upw.south.8to10 | 1 | 0.18 | 0.41 | 12 | 11 | -378.2 | -0.1 | 0.000039 | 1 | 1 | 0.496 |
| Srec.yearlings.male.juv - ST36 | 2 | 0.15 | 0.28 | 16 | 15 | -373.5 | -4.8 | 0.000234 | 2 | 4 | 0.332 |
| Srec.yearlings.male.juv - ST36 - SSTcentral.6to7 | 2 | 0.13 | 0.31 | 15 | 14 | -374.2 | -4.1 | 0.000170 | 3 | 3 | 0.388 |
| Srec.yearlings.male.juv - Yearling.surv - ST36 - SSTcentral.6to7 | 2 | 0.12 | 0.41 | 14 | 11 | -378.3 | 0.0 | 0.000015 | 4 | 2 | 0.498 |
| Srec.yearlings.male.juv - ST36 - SSTcentral.6to7 - SSTcentral.8to10 | 3 | 0.09 | 0.30 | 14 | 14 | -372.9 | -5.4 | 0.000535 | 6 | 2 | 0.401 |
| Srec.yearlings.male.juv - ST36 - ST39 - SSTcentral.6to7 | 4 | 0.11 | 0.35 | 16 | 13 | -375.4 | -2.9 | 0.000223 | 5 | 4 | 0.447 |
| Srec.yearlings.male.juv - ST36 - ST39 | 5 | 0.06 | 0.31 | 15 | 12 | -374.0 | -4.3 | 0.000664 | 7 | 3 | 0.383 |
| Srec.yearlings.male.juv - SSTcentral.6to7 | 6 | 0.11 | 0.21 | 18 | 18 | -370.5 | -7.8 | 0.003708 | 5 | 6 | 0.265 |
| Srec.yearlings.male.juv - Yearling.surv - ST36 | 6 | 0.09 | 0.33 | 17 | 12 | -375.0 | -3.3 | 0.000068 | 6 | 5 | 0.403 |
| Srec.yearlings.male.juv - ST36 - Upw.central.8to10 | 6 | 0.06 | 0.25 | 16 | 15 | -371.6 | -6.7 | 0.001136 | 7 | 4 | 0.333 |
| Srec.yearlings.male.juv - SSTsouth.8to10 - SSTcentral.6to7 | 6 | 0.09 | 0.21 | 17 | 15 | -369.8 | -8.5 | 0.007660 | 6 | 5 | 0.295 |
| Srec.yearlings.male.juv - ST36 - ST45 | 7 | 0.03 | 0.29 | 14 | 12 | -373.2 | -5.1 | 0.000284 | 10 | 2 | 0.366 |
| Srec.yearlings.male.juv - Yearling.surv - ST36 - Upw.central.8to10 | 7 | 0.04 | 0.31 | 15 | 11 | -373.4 | -4.9 | 0.000295 | 9 | 3 | 0.410 |
| Srec.yearlings.male.juv | 8 | 0.09 | 0.19 | 19 | 16 | -370.6 | -7.7 | 0.002835 | 6 | 7 | 0.218 |
| SSTsouth.8to10 - SSTcentral.6to7 | 8 | 0.09 | 0.15 | 19 | 19 | -368.2 | -10.1 | 0.026936 | 6 | 7 | 0.209 |
| Srec.yearlings.male.juv - ST36 - Upw.south.8to10 | 8 | 0.05 | 0.34 | 17 | 12 | -375.3 | -3.0 | 0.000301 | 8 | 5 | 0.408 |
| Srec.yearlings.male.juv - SSTcentral.6to7 - Upw.central.8to10 | 8 | 0.06 | 0.18 | 18 | 18 | -368.6 | -9.7 | 0.012159 | 7 | 6 | 0.266 |
| Srec.yearlings.male.juv - ST36 - SSTsouth.8to10 - SSTcentral.6to7 | 8 | 0.03 | 0.31 | 15 | 13 | -373.4 | -4.9 | 0.000640 | 10 | 3 | 0.410 |
| Srec.yearlings.male.juv - SSTsouth.8to10 - SSTcentral.6to7 - SSTnorth.8to10 | 8 | 0.05 | 0.21 | 17 | 14 | -369.0 | -9.3 | 0.014641 | 8 | 5 | 0.322 |
| SSTsouth.8to10 - SSTcentral.6to7 - SSTnorth.8to10 | 9 | 0.09 | 0.19 | 20 | 19 | -368.8 | -9.5 | 0.020712 | 6 | 8 | 0.273 |
| Srec.yearlings.male.juv | 9 | 0.03 | 0.19 | 16 | 16 | -370.6 | -7.7 | 0.002835 | 10 | 4 | 0.218 |
| variable | top20.freq |
|---|---|
| Srec.yearlings.male.juv | 18 |
| ST36 | 13 |
| SSTcentral.6to7 | 12 |
| SSTsouth.8to10 | 5 |
| Yearling.surv | 3 |
| Upw.central.8to10 | 3 |
| ST39 | 2 |
| SSTnorth.8to10 | 2 |
| Upw.south.8to10 | 2 |
| ST45 | 1 |
| SSTcentral.8to10 | 1 |
| Pup.surv | 0 |
| SSTsouth.6to7 | 0 |
| variable | weight |
|---|---|
| Srec.yearlings.male.juv | 0.28 |
| ST36 | 0.21 |
| SSTcentral.6to7 | 0.19 |
| SSTsouth.8to10 | 0.06 |
| Upw.central.8to10 | 0.06 |
| Yearling.surv | 0.04 |
| ST39 | 0.04 |
| ST45 | 0.03 |
| SSTnorth.8to10 | 0.03 |
| Upw.south.8to10 | 0.03 |
| SSTcentral.8to10 | 0.02 |
| Pup.surv | 0.01 |
| SSTsouth.6to7 | 0.00 |
Akaike weights sum to 1 ==> gives relative importance of each model, compared to the others.
| parms | Combined.rank | AdjR2.cv | AdjR2.full | FMM.cv | FMM.full | AIC.full | deltaAIC | Pval | AdjR2.rank | FMM.rank | prop.dev.explained | Akaike.weight |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Srec.yearlings.male.juv - ST36 - SSTcentral.6to7 - Upw.south.8to10 | 1 | 0.18 | 0.41 | 12 | 11 | -378.2 | -0.1 | 0.000039 | 1 | 1 | 0.496 | 0.28 |
| Srec.yearlings.male.juv - ST36 | 2 | 0.15 | 0.28 | 16 | 15 | -373.5 | -4.8 | 0.000234 | 2 | 4 | 0.332 | 0.03 |
| Srec.yearlings.male.juv - ST36 - SSTcentral.6to7 | 2 | 0.13 | 0.31 | 15 | 14 | -374.2 | -4.1 | 0.000170 | 3 | 3 | 0.388 | 0.04 |
| Srec.yearlings.male.juv - Yearling.surv - ST36 - SSTcentral.6to7 | 2 | 0.12 | 0.41 | 14 | 11 | -378.3 | 0.0 | 0.000015 | 4 | 2 | 0.498 | 0.29 |
| Srec.yearlings.male.juv - ST36 - SSTcentral.6to7 - SSTcentral.8to10 | 3 | 0.09 | 0.30 | 14 | 14 | -372.9 | -5.4 | 0.000535 | 6 | 2 | 0.401 | 0.02 |
| Srec.yearlings.male.juv - ST36 - ST39 - SSTcentral.6to7 | 4 | 0.11 | 0.35 | 16 | 13 | -375.4 | -2.9 | 0.000223 | 5 | 4 | 0.447 | 0.07 |
| Srec.yearlings.male.juv - ST36 - ST39 | 5 | 0.06 | 0.31 | 15 | 12 | -374.0 | -4.3 | 0.000664 | 7 | 3 | 0.383 | 0.03 |
| Srec.yearlings.male.juv - SSTcentral.6to7 | 6 | 0.11 | 0.21 | 18 | 18 | -370.5 | -7.8 | 0.003708 | 5 | 6 | 0.265 | 0.01 |
| Srec.yearlings.male.juv - Yearling.surv - ST36 | 6 | 0.09 | 0.33 | 17 | 12 | -375.0 | -3.3 | 0.000068 | 6 | 5 | 0.403 | 0.06 |
| Srec.yearlings.male.juv - ST36 - Upw.central.8to10 | 6 | 0.06 | 0.25 | 16 | 15 | -371.6 | -6.7 | 0.001136 | 7 | 4 | 0.333 | 0.01 |
| Srec.yearlings.male.juv - SSTsouth.8to10 - SSTcentral.6to7 | 6 | 0.09 | 0.21 | 17 | 15 | -369.8 | -8.5 | 0.007660 | 6 | 5 | 0.295 | 0.00 |
| Srec.yearlings.male.juv - ST36 - ST45 | 7 | 0.03 | 0.29 | 14 | 12 | -373.2 | -5.1 | 0.000284 | 10 | 2 | 0.366 | 0.02 |
| Srec.yearlings.male.juv - Yearling.surv - ST36 - Upw.central.8to10 | 7 | 0.04 | 0.31 | 15 | 11 | -373.4 | -4.9 | 0.000295 | 9 | 3 | 0.410 | 0.03 |
| Srec.yearlings.male.juv | 8 | 0.09 | 0.19 | 19 | 16 | -370.6 | -7.7 | 0.002835 | 6 | 7 | 0.218 | 0.01 |
| SSTsouth.8to10 - SSTcentral.6to7 | 8 | 0.09 | 0.15 | 19 | 19 | -368.2 | -10.1 | 0.026936 | 6 | 7 | 0.209 | 0.00 |
| Srec.yearlings.male.juv - ST36 - Upw.south.8to10 | 8 | 0.05 | 0.34 | 17 | 12 | -375.3 | -3.0 | 0.000301 | 8 | 5 | 0.408 | 0.07 |
| Srec.yearlings.male.juv - SSTcentral.6to7 - Upw.central.8to10 | 8 | 0.06 | 0.18 | 18 | 18 | -368.6 | -9.7 | 0.012159 | 7 | 6 | 0.266 | 0.00 |
| Srec.yearlings.male.juv - ST36 - SSTsouth.8to10 - SSTcentral.6to7 | 8 | 0.03 | 0.31 | 15 | 13 | -373.4 | -4.9 | 0.000640 | 10 | 3 | 0.410 | 0.03 |
| Srec.yearlings.male.juv - SSTsouth.8to10 - SSTcentral.6to7 - SSTnorth.8to10 | 8 | 0.05 | 0.21 | 17 | 14 | -369.0 | -9.3 | 0.014641 | 8 | 5 | 0.322 | 0.00 |
| SSTsouth.8to10 - SSTcentral.6to7 - SSTnorth.8to10 | 9 | 0.09 | 0.19 | 20 | 19 | -368.8 | -9.5 | 0.020712 | 6 | 8 | 0.273 | 0.00 |
| Srec.yearlings.male.juv | 9 | 0.03 | 0.19 | 16 | 16 | -370.6 | -7.7 | 0.002835 | 10 | 4 | 0.218 | 0.01 |
Variable importance is calculated as the sum of the Akaike weights of each model that variable appears in.
Table column “akaike.importance” is the sum of the Akaike weights of each model a certain variable appears in (so a weight of 1 means that it appears in all models).
Table column “relative.importance” is the akaike importance divided by the sum of all akaike importance values, and can be interpreted as a relative measure of importance.
| Variable | Akaike.importance | Relative.importance |
|---|---|---|
| Srec.yearlings.male.juv | 0.99 | 0.27 |
| ST36 | 0.97 | 0.27 |
| SSTcentral.6to7 | 0.73 | 0.20 |
| Yearling.surv | 0.38 | 0.10 |
| Upw.south.8to10 | 0.35 | 0.10 |
| ST39 | 0.10 | 0.03 |
| Upw.central.8to10 | 0.04 | 0.01 |
| SSTsouth.8to10 | 0.03 | 0.01 |
| ST45 | 0.02 | 0.01 |
| SSTcentral.8to10 | 0.02 | 0.01 |
| Pup.surv | 0.00 | 0.00 |
| SSTsouth.6to7 | 0.00 | 0.00 |
| SSTnorth.8to10 | 0.00 | 0.00 |
Best 3-variable models:
| parms | AdjR2.cv | FeatureMismatch.cv | Combined.rank | AdjR2.rank | FMM.rank |
|---|---|---|---|---|---|
| Srec.yearlings.male.juv - ST36 - SSTcentral.6to7 | 0.132 | 15 | 2 | 3 | 3 |
| Srec.yearlings.male.juv - ST36 - ST39 | 0.055 | 15 | 5 | 7 | 3 |
| Srec.yearlings.male.juv - Yearling.surv - ST36 | 0.086 | 17 | 6 | 6 | 5 |
| Srec.yearlings.male.juv - ST36 - Upw.central.8to10 | 0.056 | 16 | 6 | 7 | 4 |
| Srec.yearlings.male.juv - SSTsouth.8to10 - SSTcentral.6to7 | 0.086 | 17 | 6 | 6 | 5 |
Best 2-variable:
| parms | AdjR2.cv | FeatureMismatch.cv | Combined.rank | AdjR2.rank | FMM.rank |
|---|---|---|---|---|---|
| Srec.yearlings.male.juv - ST36 | 0.149 | 16 | 2 | 2 | 4 |
| Srec.yearlings.male.juv - SSTcentral.6to7 | 0.113 | 18 | 6 | 5 | 6 |
| SSTsouth.8to10 - SSTcentral.6to7 | 0.088 | 19 | 8 | 6 | 7 |
| Srec.yearlings.male.juv - Upw.central.8to10 | 0.032 | 17 | 10 | 10 | 5 |
| Srec.yearlings.male.juv - SSTnorth.8to10 | -0.012 | 19 | 15 | 14 | 7 |
Best 1-variable:
| parms | AdjR2.cv | FeatureMismatch.cv | Combined.rank | AdjR2.rank | FMM.rank |
|---|---|---|---|---|---|
| Srec.yearlings.male.juv | 0.089 | 19 | 8 | 6 | 7 |
| SSTcentral.6to7 | -0.014 | 25 | 21 | 14 | 13 |
| ST36 | -0.028 | 24 | 22 | 16 | 12 |
| SSTnorth.8to10 | -0.056 | 22 | 23 | 19 | 10 |
| Upw.central.8to10 | -0.063 | 22 | 23 | 19 | 10 |
| ST45 | -0.062 | 23 | 24 | 19 | 11 |
| Yearling.surv | -0.077 | 22 | 25 | 21 | 10 |
| Pup.surv | -0.081 | 24 | 27 | 21 | 12 |
| ST39 | -0.098 | 22 | 27 | 23 | 10 |
| SSTsouth.8to10 | -0.108 | 22 | 28 | 24 | 10 |
The top model is recreated using a glm (with a quadratic term for upwelling), with normalized variables so that the effect estimates represent the relative contribution of each variable.
(not sure how to interpret/compare the quadratic term here)
Coefficients:
## (Intercept) scale(Srec.yearlings.male.juv)
## 0.0006878489 1.6536107941
## scale(ST36) scale(SSTcentral.6to7)
## 1.3642161393 0.8125517562
## scale(Upw.south.8to10) I((scale(Upw.south.8to10))^2)
## 0.8559189110 0.8726303863
| parms | Combined.rank | AdjR2.cv | AdjR2.full | FMM.cv | FMM.full | AIC.full | deltaAIC | Pval | AdjR2.rank | FMM.rank | prop.dev.expl | Akaike.weight |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Srec.y.mj | 1 | 0.09 | 0.20 | 19 | 16 | -391.1 | 0.0 | 0.002428 | 1 | 1 | 0.224 | 0.51 |
| Srec.y.mj.fs | 2 | -0.03 | 0.14 | 21 | 17 | -388.9 | -2.2 | 0.015818 | 2 | 2 | 0.167 | 0.17 |
| Srec.y.fs.fa | 3 | -0.04 | -0.03 | 22 | 20 | -383.3 | -7.8 | 0.608846 | 3 | 3 | 0.009 | 0.01 |
| Srec.y.mj.fs.fa | 4 | -0.12 | 0.05 | 22 | 17 | -385.8 | -5.3 | 0.108769 | 5 | 3 | 0.082 | 0.04 |
| Srec.y.mj.ms.ma.fs.fa | 5 | -0.12 | 0.01 | 24 | 20 | -384.6 | -6.5 | 0.225313 | 4 | 5 | 0.048 | 0.02 |
| Srec.y.mj.ms.fs | 6 | -0.15 | 0.08 | 23 | 18 | -386.8 | -4.3 | 0.056311 | 6 | 4 | 0.110 | 0.06 |
| Srec.y.mj.ms.fs.fa | 7 | -0.16 | 0.05 | 23 | 20 | -385.9 | -5.2 | 0.095307 | 7 | 4 | 0.086 | 0.04 |
| Srec.y.mj.ms | 8 | -0.17 | 0.12 | 23 | 16 | -388.1 | -3.0 | 0.021538 | 8 | 4 | 0.147 | 0.11 |
| Srec.y.mj.ms.ma | 8 | -0.16 | 0.05 | 26 | 17 | -385.7 | -5.4 | 0.111142 | 6 | 6 | 0.079 | 0.03 |
| Srec.y.fs | 9 | -0.21 | 0.00 | 24 | 22 | -384.2 | -6.9 | 0.289152 | 9 | 5 | 0.036 | 0.02 |
## [1] "Model 1: Rank = 1, AdjR2.CV = 0.184, Feature Match = 12"
## [1] "Srec.yearlings.male.juv" "ST36"
## [3] "SSTcentral.6to7" "Upw.south.8to10"
## [1] " "
## [1] "Model 2: Rank = 2, AdjR2.CV = 0.149, Feature Match = 16"
## [1] "Srec.yearlings.male.juv" "ST36"
## [1] " "
Using the same y axis ranges for all figures, and moving the title to the bottom
## [1] "Model 3: Rank = 2, AdjR2.CV = 0.132, Feature Match = 15"
## [1] "Srec.yearlings.male.juv" "ST36"
## [3] "SSTcentral.6to7"
## [1] " "
## [1] "Model 4: Rank = 2, AdjR2.CV = 0.125, Feature Match = 14"
## [1] "Srec.yearlings.male.juv" "Yearling.surv"
## [3] "ST36" "SSTcentral.6to7"
## [1] " "
## [1] "Model 5: Rank = 3, AdjR2.CV = 0.086, Feature Match = 14"
## [1] "Srec.yearlings.male.juv" "ST36"
## [3] "SSTcentral.6to7" "SSTcentral.8to10"
## [1] " "
## [1] "Model 6: Rank = 4, AdjR2.CV = 0.109, Feature Match = 16"
## [1] "Srec.yearlings.male.juv" "ST36"
## [3] "ST39" "SSTcentral.6to7"
## [1] " "
## [1] "Model 7: Rank = 5, AdjR2.CV = 0.055, Feature Match = 15"
## [1] "Srec.yearlings.male.juv" "ST36"
## [3] "ST39"
## [1] " "
## [1] "Model 8: Rank = 6, AdjR2.CV = 0.113, Feature Match = 18"
## [1] "Srec.yearlings.male.juv" "SSTcentral.6to7"
## [1] " "
## [1] "Model 9: Rank = 6, AdjR2.CV = 0.086, Feature Match = 17"
## [1] "Srec.yearlings.male.juv" "Yearling.surv"
## [3] "ST36"
## [1] " "
## [1] "Model 10: Rank = 6, AdjR2.CV = 0.056, Feature Match = 16"
## [1] "Srec.yearlings.male.juv" "ST36"
## [3] "Upw.central.8to10"
## [1] " "
## [1] "Model 17: Rank = 8, AdjR2.CV = 0.062, Feature Match = 18"
## [1] "Srec.yearlings.male.juv" "SSTcentral.6to7"
## [3] "Upw.central.8to10"
## [1] " "
The predictions of the top 20 models are averages according to Akaike weights.
Model averaging stats:
Adjusted R2: 0.4173168
Feature mismatches: 12
% deviance explained: 0.5005573
| Srec.method | FMM | AdjR2 | Prop.dev.explained | AIC |
|---|---|---|---|---|
| S.smooth | 17 | 0.15 | 0.187 | -369.3 |
| S.observed | 16 | 0.18 | 0.218 | -370.6 |
| Srec.method | FMM | AdjR2 | Prop.dev.explained | AIC |
|---|---|---|---|---|
| S.smooth | 16 | 0.40 | 0.278 | -291.2 |
| S.observed | 15 | 0.43 | 0.313 | -292.4 |
## Var1 Freq
## 1 1984-02 1
## 2 1984-05 1
## 3 1984-07 11
## 4 1984-08 36
## 5 1984-09 27
## 6 1984-10 19
## year N
## 1 1984 66388.13
## 2 1985 64430.63
## 3 1986 66599.83
## 4 1987 70652.75
## 5 1988 76592.43
## 6 1989 82278.99
(slightly different values because scaling is now done for some variables on the 1984 - 2014 data instead of the 1984 - 2012 data)
Outbreak size (circle size) = cases/N.
Outbreak category:
- no outbreak < 0.0004 cases/N
- outbreak > 0.0004 cases/N
(this corresponds with the existing outbreak categories, where no outbreak is the same category, while outbreak is medium + large outbreaks.)
Susceptibles = susceptible anomaly.
Environmental component = anomaly of the linear combination of the environmental variables, with coefficients determined by model fit.
Same data but separate time series:
We already know that survival correlates with SST, from DeLong et al 2017.
But they have not used the exact same variables we did, so in order to assess how the top environmental variables that were selected in the lepto outbreak size model correlate with survival, we here do a similar analysis.
We also do this for the correlation between reconstructed susceptibles and environmental variables.
Only the three environmental variables in the top models are considered:
- ST36 - SST central Jun-Jul - Upwelling south Aug-Oct
As for the cases models, we first select the best degree of nonlinearity (max 1 knot), but now using AIC.
##
## Call:
## lm(formula = Srec.yearlings.male.juv ~ ST36, data = d1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.3647 -0.4086 0.1339 0.7385 1.3113
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.003922 0.185726 0.021 0.983
## ST36 -0.190046 0.189735 -1.002 0.325
##
## Residual standard error: 0.9999 on 27 degrees of freedom
## Multiple R-squared: 0.03583, Adjusted R-squared: 0.0001174
## F-statistic: 1.003 on 1 and 27 DF, p-value: 0.3254
## Analysis of Variance Table
##
## Response: Srec.yearlings.male.juv
## Df Sum Sq Mean Sq F value Pr(>F)
## ST36 1 1.0032 1.00317 1.0033 0.3254
## Residuals 27 26.9968 0.99988
##
## Call:
## lm(formula = Srec.yearlings.male.juv ~ SSTcentral.6to7, data = d1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.4117 -0.5908 0.1693 0.6589 1.4948
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.02785 0.17867 -0.156 0.8773
## SSTcentral.6to7 -0.46318 0.24902 -1.860 0.0738 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9588 on 27 degrees of freedom
## Multiple R-squared: 0.1136, Adjusted R-squared: 0.08075
## F-statistic: 3.46 on 1 and 27 DF, p-value: 0.07381
## Analysis of Variance Table
##
## Response: Srec.yearlings.male.juv
## Df Sum Sq Mean Sq F value Pr(>F)
## SSTcentral.6to7 1 3.1802 3.1802 3.4596 0.07381 .
## Residuals 27 24.8198 0.9193
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Call:
## lm(formula = Srec.yearlings.male.juv ~ Upw.south.8to10, data = d1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.2078 -0.6201 0.1837 0.9015 1.4348
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.617e-16 1.890e-01 0.000 1.000
## Upw.south.8to10 -3.886e-02 1.923e-01 -0.202 0.841
##
## Residual standard error: 1.018 on 27 degrees of freedom
## Multiple R-squared: 0.00151, Adjusted R-squared: -0.03547
## F-statistic: 0.04083 on 1 and 27 DF, p-value: 0.8414
## Analysis of Variance Table
##
## Response: Srec.yearlings.male.juv
## Df Sum Sq Mean Sq F value Pr(>F)
## Upw.south.8to10 1 0.0423 0.04228 0.0408 0.8414
## Residuals 27 27.9577 1.03547
## Analysis of Deviance Table
##
## Model 1: Yearling.surv ~ 1 + ST36
## Model 2: Yearling.surv ~ 1
## Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1 27 1.618
## 2 28 2.016 -1 -0.39798 0.5281
## Analysis of Deviance Table
##
## Model 1: Yearling.surv ~ 1 + SSTcentral.6to7
## Model 2: Yearling.surv ~ 1
## Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1 27 1.9394
## 2 28 2.0160 -1 -0.076633 0.7819
## Analysis of Deviance Table
##
## Model 1: Yearling.surv ~ 1 + Upw.south.8to10
## Model 2: Yearling.surv ~ 1
## Resid. Df Resid. Dev Df Deviance Pr(>Chi)
## 1 27 1.878
## 2 28 2.016 -1 -0.13795 0.7103
# Susceptibles vs cases
S vs cases vs time 3D
S vs cases vs time all data
Green = low, red = high.
! Note: axis Cases/N goes from large to small in the default 3D plot setting
Green = low, red = high.
! Note: axis Cases/N goes from large to small in the default 3D plot setting
Green = low, red = high.
! Note: axis Cases/N goes from large to small in the default 3D plot setting
Green = low, red = high.
! Note: axis Cases/N goes from large to small in the default 3D plot setting
Exponential growth rates calculated for outbreak years.
Using Poisson regression to estimate the exponential growth rate.
Fitted time periods for each year = beginning of June until the beginning of the month with the maximum number of cases (the peak month).
There is no significant correlation between population size and growth rate (pos or neg).
## Analysis of Variance Table
##
## Response: growth.rate$lambda
## Df Sum Sq Mean Sq F value Pr(>F)
## growth.rate$N 1 0.046913 0.046913 1.8793 0.1977
## Residuals 11 0.274596 0.024963
Considering the fact that none of the distributions result in good residuals, and those for binomial and gamma-log families are the same, it might be informative to do lassa regression assuming a binomial distribution.
## [1] "(Intercept)" "Srec.yearlings.male.juv"
## [3] "ST36"
## [1] "(Intercept)" "Srec.yearlings.male.juv"
## [3] "Pup.surv" "ST36"
## [1] "(Intercept)" "Srec.yearlings.male.juv"
## [3] "Pup.surv" "ST36"
## [5] "ST45" "SSTsouth.8to10"
## [1] "(Intercept)" "Srec.yearlings.male.juv"
## [3] "Pup.surv" "Yearling.surv"
## [5] "ST36" "ST45"
## [7] "SSTsouth.8to10"
## var rel.inf
## Srec.yearlings.male.juv Srec.yearlings.male.juv 24.2060898
## Upw.central.8to10 Upw.central.8to10 20.1672833
## ST36 ST36 18.5638816
## Yearling.surv Yearling.surv 13.6927443
## SSTcentral.6to7 SSTcentral.6to7 8.2878982
## SSTsouth.6to7 SSTsouth.6to7 4.2126872
## Pup.surv Pup.surv 4.1833724
## ST45 ST45 1.8401817
## SSTsouth.8to10 SSTsouth.8to10 1.4135953
## ST39 ST39 1.3685602
## Upw.south.8to10 Upw.south.8to10 1.3183986
## SSTnorth.8to10 SSTnorth.8to10 0.7453075
## SSTcentral.8to10 SSTcentral.8to10 0.0000000
None of the density distributions below result in good residuals.
Gamma log results in stronger correlations.
No difference in AIC between different link functions for Gamma distribution.
==> I think we’re good with Gamma log.
##
## Call:
## glm(formula = caseN ~ Srec.yearlings.male.juv + SSTsouth.8to10,
## family = "binomial", data = d1)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.034578 -0.015572 -0.002384 0.008084 0.040742
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -7.3118 7.6002 -0.962 0.336
## Srec.yearlings.male.juv 0.4328 8.3597 0.052 0.959
## SSTsouth.8to10 0.1334 9.0662 0.015 0.988
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 0.0132854 on 28 degrees of freedom
## Residual deviance: 0.0099502 on 26 degrees of freedom
## AIC: 6.0415
##
## Number of Fisher Scoring iterations: 10
##
## Call:
## glm(formula = caseN ~ Srec.yearlings.male.juv + SSTsouth.8to10,
## family = Gamma(link = "log"), data = d1)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.4401 -0.8183 -0.1542 0.3873 1.3518
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -7.31778 0.13846 -52.852 < 2e-16 ***
## Srec.yearlings.male.juv 0.40896 0.13869 2.949 0.00666 **
## SSTsouth.8to10 0.02706 0.18324 0.148 0.88374
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for Gamma family taken to be 0.5379045)
##
## Null deviance: 20.981 on 28 degrees of freedom
## Residual deviance: 16.391 on 26 degrees of freedom
## AIC: -364.58
##
## Number of Fisher Scoring iterations: 6
##
## Call:
## glm(formula = caseN ~ Srec.yearlings.male.juv + SSTsouth.8to10,
## family = Gamma(link = "inverse"), data = d1)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.4357 -0.7443 -0.1609 0.2296 1.4716
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1629.0 248.4 6.559 5.92e-07 ***
## Srec.yearlings.male.juv -652.4 260.9 -2.501 0.019 *
## SSTsouth.8to10 -102.9 218.4 -0.471 0.642
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for Gamma family taken to be 0.5807588)
##
## Null deviance: 20.981 on 28 degrees of freedom
## Residual deviance: 16.161 on 26 degrees of freedom
## AIC: -365.03
##
## Number of Fisher Scoring iterations: 6
##
## Call:
## glm(formula = caseN ~ Srec.yearlings.male.juv + SSTsouth.8to10,
## family = Gamma(link = "identity"), data = d1)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.5192 -0.8636 -0.1328 0.4677 1.2175
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.063e-04 9.772e-05 7.228 1.12e-07 ***
## Srec.yearlings.male.juv 2.423e-04 6.959e-05 3.482 0.00178 **
## SSTsouth.8to10 -6.787e-05 9.514e-05 -0.713 0.48199
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for Gamma family taken to be 0.5053988)
##
## Null deviance: 20.981 on 28 degrees of freedom
## Residual deviance: 16.580 on 26 degrees of freedom
## AIC: -364.22
##
## Number of Fisher Scoring iterations: 25